1. Field of the Invention
This invention relates to an improved apparatus and method for measuring electrical conductivity or resistivity in liquids, and more particularly, it relates to a conductivity measuring apparatus having a phase-synchronous rectifier and integrator for accurately measuring electrical conductivity in the presence of parasitic capacitance.
2. Background of the Invention
The measurement of conductivity or resistivity is desired in a number of applications. For example, in the pharmaceutical industry, many processes require the use of ultra-pure water. Conductivity measurements yield an indication of ionic concentrations in water. Ultra-pure water has a conductivity below a given level. Conductivity measurements can therefore be used to reliably and accurately determine the purity of water.
Another application where it is desirable to measure the conductivity of a liquid is the determination of the concentration of total dissolved solids in water. For example, investigators desiring to determine the level of pollution in river water want to know the concentration of total dissolved solids in the water. The higher the concentration of total dissolved solids in the water, the higher the conductivity, or inversely, the lower the resistivity of the water.
The volume conductivity, or just "conductivity", is defined as the conductance of one cubic centimeter of a liquid at a specific temperature. Conductivity is typically measured in mhos/cm ({character pullout}/cm), or Siemens/cm (S/cm), and micro mhos/cm (.mu.{character pullout}/cm), or micro Siemens/cm (.mu.S/cm). Ultra-pure water typically has a conductivity of 0.2 micro mhos/cm or less. Volume resistivity ("resistivity") is the inverse of conductivity and is typically measured in ohm-cm (.OMEGA.-cm), or megohm-cm (M.OMEGA.-cm). Ultra-pure water typically has a resistivity of 5 megohms-cm or greater.
Conductivity of a liquid is typically measured by immersing two electrodes contained in a conductivity cell in the liquid, applying an excitation to the liquid, and measuring the resultant voltage v.sub.c between the electrodes and the current i.sub.c flowing through the electrodes. Because a direct current ("DC") excitation can cause ions present in the liquid to migrate to the electrodes, interfering with the conductivity measurement, an alternating current ("AC") excitation of sufficiently low amplitude and sufficiently high frequency is often used.
Measurements of conductivity and resistivity vary depending upon the cell used to make the measurements, the temperature of the liquid being measured, and the concentration of ions or other electrically conductive material in the liquid. A cell with fixed dimensions and configuration is typically used. For a given fixed cell, a cell constant K may be defined as a function of conductive cell surface area and conductive path length For a cell with two flat parallel plates of area A and separation distance L, the cell constant K is found to be the length L of the conductive path between the electrodes, divided by the conducting area A of the electrodes, so that K=L/A.
For a given cell, the conductivity and resistivity of a liquid are then given by ##EQU1##
Where:
i.sub.c =the electric current flowing between the cell electrodes, in Amperes, PA1 v.sub.c =the voltage across the cell electrodes, in Volts, and PA1 K=the cell constant, in cm.sup.-1.
The conductivity cell immersed in liquid may be electrically modeled as a resistor R.sub.c with value equal to the resistivity .rho. times the cell constant K, such that EQU R.sub.c =K.rho.. (eq.3)
However, accurate measurement of R.sub.c is difficult when an AC excitation is used due to capacitive effects of the cell, as well as capacitive effects of the lead wires to the cell. At the interface between each cell electrode and the liquid is a series capacitance C.sub.S. Between each electrode is a capacitance in parallel with the resistance R.sub.c, represented by C.sub.p. Including the capacitive effects, the cell may be electrically modeled as a parallel capacitance C.sub.p in parallel with liquid resistance R.sub.c, both in series connection with series capacitance C.sub.s, as shown in FIG. 1. Lead wire cabling capacitance (not shown) would appear as a capacitance in parallel across the circuit of FIG. 1.
The cell capacitances C.sub.p and C.sub.s exhibit impedances to an AC excitation which vary inversely as a function of the excitation frequency f (measured in cycles/second or Hertz). For a relatively low frequency f, the impedances of C.sub.p and C.sub.S can be quite large for fixed values of C.sub.p and C.sub.s. For small values of R.sub.c, the impedance of series capacitance C.sub.s can be large compared to R.sub.c, thus giving rise to an erroneously large measured value for R.sub.c. For large values of R.sub.c, the impedance of parallel capacitance C.sub.p can be small relative to R.sub.c, thus giving rise to erroneously small measured values of R.sub.c.
For a relatively high excitation frequency f the impedances of C.sub.p and C.sub.s can be quite small for fixed values of C.sub.p and C.sub.s. For small values of R.sub.c, the impedance of series capacitance C.sub.s can be large compared to R.sub.c, thus giving rise to an erroneously large measured value for R.sub.c. For large values of R.sub.c, the impedance of parallel capacitance C.sub.p can be small relative to R.sub.c, thus giving rise to an erroneously small measured value for R.sub.c.
In general, at a given frequency f and fixed C.sub.s and C.sub.p, as the resistivity of a sampled liquid increases, the impedance due to series capacitance C.sub.s can be ignored, while the impedance of parallel capacitance C.sub.p causes an erroneously small value for R.sub.c to be measured. Conversely, as the resistivity of a sampled liquid decreases, the impedance due to parallel capacitance C.sub.p can be ignored, while the impedance of series capacitance C.sub.s causes an erroneously large value for R.sub.c to be measured.
Thus, for large values of cell resistance R.sub.c, the measurement error is largely due to the presence of parallel capacitance C.sub.p. For small values of cell resistance R.sub.c, the primary source of measurement error is due to the presence of series capacitance C.sub.s.
Various efforts to measure conductivity in the presence of capacitive effects are known in the prior art. An early method uses an AC conductance bridge, wherein different reactances are inserted into the arms of the bridge to compensate either or both C.sub.s and C.sub.p. While this method is effective, it is generally slow and not easily automated.
Another measurement technique uses square-wave excitation and center-sampling of the voltage waveform across the cell. The parallel capacitance is charged to saturation during the first part of the square-wave cycle. The cell voltage is then sampled during a later portion of the cycle during which the series capacitance is charging in a linear fashion. The value of the series capacitance can be determined from the rate of charge of the capacitance and mathematically subtracted from the output based on the cell voltage to determine the cell resistance. This measurement technique suffers from the disadvantage of relying upon the use of a second-order polynomial to approximate the amount of measurement error. Thus, this prior art method does not eliminate the source of error itself, the voltage due to the series capacitance.